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Title: On the formulas for Pi(​x) and Psi(x) of Riemann and von Mangoldt
Author(s): Kunik, MatthiasLook up in the Integrated Authority File of the German National Library
Issue Date: 2023
Extent: 1 Online-Ressource (42 Seiten)
Type: Preprint
Language: English
Publisher: Universitätsbibliothek, Magdeburg
URN: urn:nbn:de:gbv:ma9:1-1981185920-1108703
Subjects: Explicit formulas
Distribution of prime numbers
Riemann zeta function
Fourier analysis
Abstract: Using the Mellin transform and the complex exponential integral we derive various representation formulas for the factors of the entire functions in Hadamards product theorem. The application of these results on Riemann’s zeta function leads to a new derivation of Rie- mann’s prime number formula for Pi(x). We will thereby present a correct version of this formula, which is given in a wrong way in the literature. Using the nontrivial zeros of the Zeta function we also obtain explicit formulas for regularizations of von Mangoldt’s function Psi(x). These regularizations are based on cardinal B-splines and Gaussian integration kernels, which are related by the Central Limit Theorem. Our results will then be generalized to a windowed Mellin or Fourier transform with a Gaussian window function.
Open Access: Open access publication
License: (CC BY-SA 4.0) Creative Commons Attribution ShareAlike 4.0(CC BY-SA 4.0) Creative Commons Attribution ShareAlike 4.0
Appears in Collections:Fakultät für Mathematik (OA)

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